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                  <li class="toctree-l1"><a class="reference internal" href="#">计算机组成原理实验</a>
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                <li class="toctree-l2"><a class="reference internal" href="../../%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86%E5%AE%9E%E9%AA%8C/3/3/">MIPS指令集1</a>
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                  <li class="toctree-l1 current"><a class="reference internal current" href="#">概率论</a>
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                <li class="toctree-l2"><a class="reference internal" href="../5.%20%E5%A4%A7%E6%95%B0%E5%AE%9A%E5%BE%8B%E5%8F%8A%E4%B8%AD%E5%BF%83%E6%9E%81%E9%99%90%E5%AE%9A%E7%90%86/">大数定律及中心极限定理</a>
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                <li class="toctree-l2 current"><a class="reference internal current" href="./">样本及抽样分布</a>
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    <li class="toctree-l3"><a class="reference internal" href="#_2">随机样本</a>
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    <li class="toctree-l3"><a class="reference internal" href="#_3">直方图和箱线图</a>
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    <li class="toctree-l4"><a class="reference internal" href="#_5">统计量</a>
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    <li class="toctree-l4"><a class="reference internal" href="#chi2">\(\chi^2\)分布</a>
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    <li class="toctree-l4"><a class="reference internal" href="#t">t分布</a>
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    <li class="toctree-l4"><a class="reference internal" href="#f">F分布</a>
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    <li class="toctree-l4"><a class="reference internal" href="#_7">正态总体的样本均值与样本方差的分布</a>
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                <li class="toctree-l2"><a class="reference internal" href="../8.%20%E5%81%87%E8%AE%BE%E9%AA%8C%E8%AF%81/">假设验证</a>
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                <h1 id="_1">样本及抽样分布</h1>
<h2 id="_2">随机样本</h2>
<p><strong>总体</strong>：试验的全部可能的观测值<br />
<strong>个体</strong>：一个可能的观察值<br />
<strong>容量</strong>：总体中包含的个体的个数  <br />
根据容量是否有限可划分<strong>有限总体</strong>和<strong>无限总体</strong><br />
<strong>样本</strong>：从总体中抽取若干个体观察，这部分被抽取的个体称为样本，抽取数量称为样本容量<br />
更严谨的定义：设X是具有分布函数F的随机变量，若<span class="arithmatex">\(X_1,X_2,\cdots,X_n\)</span>是具有同一分布函数F的相互独立的随机变量，则称<span class="arithmatex">\(X_1,X_2,\cdots,X_n\)</span>为从分布
函数F（或总体F、或总体X）得到的<strong>容量为n的简单样本</strong>，简称<strong>样本</strong>，他们的观测值<span class="arithmatex">\(x_1,x_2,\cdots,x_n\)</span>称为<strong>样本值</strong>，又被称为X的n个<strong>独立的观测值</strong></p>
<h2 id="_3">直方图和箱线图</h2>
<p>直方图和箱线图都是用于整理数据的工具</p>
<p><strong>频率直方图</strong>：取一个包含所有数据的区间，将区间均分为n等份，每份长度称为<strong>组距</strong><span class="arithmatex">\(\Delta\)</span>，小区间的端点称为<strong>组限</strong>。然后统计落在各区间的数据个数，并在xoy平面上绘制成不同高度的等宽矩形，高度即y轴坐标等于<span class="arithmatex">\(\frac{f_i}{n}/\Delta\)</span>，其中<span class="arithmatex">\(f_i\)</span>为频数。当n很大时，由于频率接近于概率，因此一般来说频率直方图的外廓曲线接近于总体X的概率密度曲线</p>
<p><strong>样本分位数</strong>：设有容量为n的样本观察值<span class="arithmatex">\(x_1,x_2,\cdots,x_n\)</span>，样本p分位数(0&lt;p&lt;1)记为<span class="arithmatex">\(x_p\)</span>，它具有以下性质：（1）至少有np个观察值小于或等于<span class="arithmatex">\(x_p\)</span>（2）至少有n(1-p)个观察值大于或等于<span class="arithmatex">\(x_p\)</span>。具体来说，若np不是整数，则样本p分位数为<span class="arithmatex">\(x_{([np]+1)}\)</span>；若np是整数，则样本p分位数为<span class="arithmatex">\(\frac{1}{2}[x_{(np)}+x_{(np+1)}]\)</span>。当p取0.25、0.5、0.75时，样本p分位数分别称为<strong>第一四分位数</strong><span class="arithmatex">\(Q_1\)</span>、<strong>样本中位数</strong><span class="arithmatex">\(Q_2或M\)</span>、<strong>第三四分位数</strong><span class="arithmatex">\(Q_3\)</span></p>
<p><strong>箱线图</strong>：记最大值为Max最小值为Min，具体画法如下。箱线图可以反映数据的中心位置、散布程度（区间越短越密集，越长越分散）、对称性。
<img alt="" src="../img/1.jpeg" /></p>
<h2 id="_4">抽样分布</h2>
<h3 id="_5">统计量</h3>
<p>设<span class="arithmatex">\(X_1,X_2,\cdots,X_n\)</span>是来自总体X的一个样本，<span class="arithmatex">\(g(X_1,X_2,\cdots,X_n)\)</span>是<span class="arithmatex">\(X_1,X_2,\cdots,X_n\)</span>的函数，若g中<em>不含未知参数</em>，则称<span class="arithmatex">\(g(X_1,X_2,\cdots,X_n)\)</span>是一个<strong>统计量</strong>。统计量是一个随机变量</p>
<p>常见统计量：</p>
<ol>
<li><strong>样本均值</strong>：<span class="arithmatex">\(\bar X=\frac{1}{n}\sum_{i=1}^nX_i\)</span></li>
<li><strong>样本方差</strong>：<span class="arithmatex">\(S^2=\frac{1}{n-1}\sum_{i=1}^n(X_i-\bar X)^2=\frac{1}{n-1}(\sum_{i=1}^n X_i^2 - n\bar X^2)\)</span></li>
<li><strong>样本标准差</strong>：<span class="arithmatex">\(S=\sqrt{S^2}\)</span></li>
<li>
<p><strong>样本k阶（原点）矩</strong>：<span class="arithmatex">\(A_k=\frac{1}{n}\sum_{i=1}^nX_i^k,k=1,2,\cdots\)</span></p>
<blockquote>
<p>当<span class="arithmatex">\(n\to \infty\)</span>时，<span class="arithmatex">\(A_k\overset{P}{\rightarrow}\mu_k\)</span></p>
</blockquote>
</li>
<li>
<p><strong>样本k阶中心矩</strong>：<span class="arithmatex">\(B_k=\frac{1}{n}\sum_{i=1}^n(X_i-\bar X)^k,k=2,3,\cdots\)</span>
这些统计值的观测值分别记为<span class="arithmatex">\(\bar x,s^2,s,a_k,b_k\)</span>，均可类似地由样本观察值算出</p>
</li>
</ol>
<h3 id="_6">经验分布函数</h3>
<p>设<span class="arithmatex">\(x_1,x_2\cdots,x_n\)</span>是来自分布函数为F(x)的总体X的样本观察值的经验分布函数，记为<span class="arithmatex">\(F_n(x)\)</span>，定义为样本观察值<span class="arithmatex">\(x_1,x_2\cdots,x_n\)</span>中小于或等于指定值x所占的比例，即</p>
<div class="arithmatex">\[F_n(x)=\frac{\#(x_i\leq x)}{n},-\infty&lt;x&lt;\infty\]</div>
<p>其中<span class="arithmatex">\(\#(x_i\leq x)\)</span>表示<span class="arithmatex">\(x_1,x_2\cdots,x_n\)</span>中小于或等于x的个数。</p>
<p>不难发现<span class="arithmatex">\(F_n\)</span>是分布函数。</p>
<p>设<span class="arithmatex">\(X_1,X_2\cdots,X_n\)</span>是来自以F(x)为分布函数的总体X的样本，<span class="arithmatex">\(F_n(x)\)</span>是经验分布函数，则有</p>
<div class="arithmatex">\[P\{\lim_{n\to\infty}\sup_{-\infty&lt;x&lt;\infty}|F_n(x)-F(x)|=0\}=1\]</div>
<p>其中sup是上确界的意思。
也就是说当n充分大时，经验分布函数能很好地逼近分布函数</p>
<h3 id="chi2"><span class="arithmatex">\(\chi^2\)</span>分布</h3>
<p><strong>伽马函数</strong>：</p>
<div class="arithmatex">\[\Gamma(\alpha)=\int_0^\infty x^{\alpha-1}e^{-x}dx\]</div>
<p>其具有以下性质：</p>
<ol>
<li>对<span class="arithmatex">\(\alpha\geq1，\Gamma(\alpha+1)=\alpha\Gamma(\alpha)\)</span></li>
<li>对正整数n，<span class="arithmatex">\(\Gamma(n)=(n-1)!\)</span></li>
<li><span class="arithmatex">\(\Gamma(\frac{1}{2})=\pi^{\frac{1}{2}}\)</span></li>
</ol>
<p><strong>伽马分布</strong>：
若随机变量X服从参数为<span class="arithmatex">\(\alpha&gt;0,\theta&gt;0\)</span>的伽马分布，则记为<span class="arithmatex">\(X\sim \Gamma(\alpha,\theta)\)</span>，其概率密度为</p>
<div class="arithmatex">\[f(x)=\left\{\begin{aligned}&amp;\frac{1}{\theta^\alpha\Gamma(\alpha)}x^{\alpha-1}e^{-x/\theta},&amp;y&gt;0,\\&amp;0,&amp;其他.\end{aligned}\right.\]</div>
<p>若<span class="arithmatex">\(X\sim \Gamma(\alpha,\theta),Y\sim \Gamma(\beta,\theta)\)</span>，则<span class="arithmatex">\(X+Y\sim\Gamma(\alpha+\beta,\theta)\)</span>。这可以推广到任意多伽马分布的加和</p>
<p><span class="arithmatex">\(\chi^2\)</span><strong>分布</strong>
设<span class="arithmatex">\(X_1,X_2,\cdots,X_n\)</span>是来自总体N(0,1)的样本，则称统计量<span class="arithmatex">\(\chi^2=X_1^2+X_2^2+\cdots+X_n^2\)</span>为服从自由度为n的<span class="arithmatex">\(\chi^2\)</span><strong>分布</strong>，记为<span class="arithmatex">\(\chi^2\sim\chi^2(n)\)</span>，其概率密度为</p>
<div class="arithmatex">\[f(y)=\left\{\begin{aligned}&amp;\frac{1}{2^{n/2}\Gamma(n/2)}y^{n/2-1}e^{-y/2},&amp;y&gt;0,\\&amp;0,&amp;其他.\end{aligned}\right.\]</div>
<p>可以发现，服从自由度为n的<span class="arithmatex">\(\chi^2\sim\Gamma(\frac{n}{2},2)\)</span></p>
<p>卡方分布满足可加性：若<span class="arithmatex">\(\chi_1^2\sim\chi^2(n_1),\chi_2^2\sim\chi^2(n_2)\)</span>，则<span class="arithmatex">\(\chi_1^2+\chi_2^2\sim\chi^2(n_1+n_2)\)</span></p>
<p>卡方分布的期望为n，方差为2n。</p>
<p>设<span class="arithmatex">\(\chi_\alpha^2(n)\)</span>是<span class="arithmatex">\(\chi^2(n)\)</span>的上<span class="arithmatex">\(\alpha\)</span>分位数，当n充分大时，<span class="arithmatex">\(\chi_\alpha^2(n)\approx\frac{1}{2}(z_\alpha+\sqrt{2n-1})^2\)</span>，其中<span class="arithmatex">\(z_\alpha\)</span>是标准正态分布的上<span class="arithmatex">\(\alpha\)</span>分位数</p>
<h3 id="t">t分布</h3>
<p>设<span class="arithmatex">\(X\sim N(0,1),Y\sim \chi^2(n)\)</span>，且X,Y相互独立，则称随机变量</p>
<div class="arithmatex">\[t=\frac{X}{\sqrt{Y/n}}\]</div>
<p>服从自由度为n的<strong>t分布</strong>，或称为学生氏分布，记为<span class="arithmatex">\(t\sim t(n)\)</span>，其概率密度为</p>
<div class="arithmatex">\[h(t)=\frac{\Gamma[(n+1)/2]}{\sqrt{\pi n}\Gamma(n/2)}\left(1+\frac{t^2}{n}\right)^{-(n+1)/2},-\infty&lt;t&lt;\infty\]</div>
<p>t分布相对y轴对称。当n足够大时t分布近似于标准正态分布。</p>
<h3 id="f">F分布</h3>
<p>设<span class="arithmatex">\(U\sim \chi^2(n_1),V\sim \chi^2(n_2)\)</span>，且U,V相互独立，则称随机变量</p>
<div class="arithmatex">\[F=\frac{U/n_1}{V/n_2}\]</div>
<p>服从自由度为<span class="arithmatex">\((n_1,n_2)\)</span>的<strong>F分布</strong>，记为<span class="arithmatex">\(F\sim F(n_1,n_2)\)</span>，其概率密度为</p>
<div class="arithmatex">\[\psi(y)=\left\{\begin{aligned}&amp;\frac{\Gamma[(n_1+n_2)/2](n_1/n_2)^{n_1/2}y^{(n_1/2)-1}}{\Gamma(n_1/2)\Gamma(n_2/2)[1+(n_1y/n_2)]^{(n_1+n_2)/2}},&amp;y&gt;0,\\&amp;0,&amp;其他.\end{aligned}\right.\]</div>
<p>可以证明<span class="arithmatex">\(\frac{1}{F}\sim F(n_2,n_1),F_{1-\alpha}(n_1,n_2)=\frac{1}{F_\alpha(n_2,n_1)}\)</span></p>
<h3 id="_7">正态总体的样本均值与样本方差的分布</h3>
<p>设<span class="arithmatex">\(X_1,X_2,\cdots,X_n\)</span>是来自正态总体<span class="arithmatex">\(N(\mu,\sigma^2)\)</span>的样本，<span class="arithmatex">\(\bar X\)</span>是样本均值,<span class="arithmatex">\(S^2\)</span>是样本方差，则有</p>
<ol>
<li><span class="arithmatex">\(\bar X\sim N(\mu, \sigma^2/n)\)</span></li>
<li><span class="arithmatex">\(\frac{(n-1)S^2}{\sigma^2}\sim \chi^2(n-1)\)</span></li>
<li>样本均值和样本方差相互独立</li>
<li><span class="arithmatex">\(\frac{\bar X-\mu}{S/\sqrt n}\sim t(n-1)\)</span></li>
</ol>
<p>设<span class="arithmatex">\(X_1,X_2,\cdots,X_{n_1}\)</span>是来自正态总体<span class="arithmatex">\(N(\mu_1,\sigma_1^2)\)</span>的样本，<span class="arithmatex">\(Y_1,Y_2,\cdots,Y_{n_2}\)</span>是来自正态总体<span class="arithmatex">\(N(\mu_2,\sigma_2^2)\)</span>的样本，<span class="arithmatex">\(\bar X,\bar Y\)</span>分别是X和Y的样本均值,<span class="arithmatex">\(S_1^2, S_2^2\)</span>分别是X和Y的样本方差，则有</p>
<ol>
<li><span class="arithmatex">\(\frac{S_1^2/S_2^2}{\sigma_1^2/\sigma_2^2}\sim F(n_1-1,n_2-1)\)</span></li>
<li>当<span class="arithmatex">\(\sigma_1^2=\sigma_2^2=\sigma^2\)</span>时</li>
</ol>
<div class="arithmatex">\[\frac{(\bar X-\bar Y)-(\mu_1-\mu_2)}{S_w\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}}\sim t(n_1+n_2-2)\]</div>
<p>其中<span class="arithmatex">\(S_w^2=\frac{(n_1-1)S_1^2+(n_2-1)S_2^2}{n_1+n_2-2},S_w=\sqrt{S_w^2}\)</span></p>
              
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